I have seen that the TDR measurement setup has the following math:

FFT of the derivative of CH3 – FFT of the derivative of CH4

Although, I cannot find the math explanation for this in links and so the bibliography.

Can you please share some insights or articles?

I haven’t ever derived or really discussed the equation, though I have shown it and discussed the method frequently. If you look at this 70GHz image, taken by Mike Martin for us, you can see the perfect edge of our J2151A TDR. We even trademarked PerfectPulse.

The other 2 images show the dVdt of the edge. This is a bandwidth limited impulse. The bandwidth is 0.35/edge speed.

Looking at the FFT of an impulse it is flat response up to this bandwidth. You can see that in the FFT.

This is of course only looking at the 1 channel that is the generator. Through probes, cables, etc, this edge can be degraded, but we can measure what is at this “port”. If I connected this to a second channel through a perfect cable, the second channel would see the same thing as the first and dividing them would provide the transfer function. If we present the signal to the DUT through 50 Ohms (our TDR splitter does) we’ll measure the S-parameter, S21. The second channel also looks at the derivative, since it is also a response of the signal comb, though the scope connects these points with smoothing for us. This is a division, but since the measurements are in dB we subtract these to get the division.

This is a neat trick. We can use this to measure uWave frequency response. I attached a few measurements using this “VNA”. You’ll see 1.57GHz and 4GHz bandpass filters and also a 20dB attenuator for verification of the math.

If we turn the splitter around so we DON’T present 50 Ohms, we can also measure the transfer function of probes, cables, etc without the 50 Ohm damping. See the attached PML probe measurement and the Tek005 (power rail probe).

Of course this also works to measure the scope transfer function and the bandwidth limiters and the step vs flat response selections, etc. It’s quite a handy tool that fits in your pocket. We do have setup files for all of these, since we showed them in our Tektronix events.

FYI, this is also how the calibration labs validate our 10.5GHz specification for our TDR. Using the same method as I showed here. It just requires a scope with much more bandwidth than our 10.5GHz edge. Using the signal generator for TDR, just looks at the same signal, but a bit differently, evaluating mostly the reflections.

One more highlight. Looking at the attached Tek003_001, you can see ringing on the edges. Both edges. Interestingly, you can also see the ringing BEFORE the transition from low to hi and hi to low. Hmmm, that says there is a pre-response or a non-causal response. How can that be? That is a perfect picture of the Gibbs phenomenon 😉

So much fun in one pocket sized gadget…

Also see this helpful article

Measuring a Scope Probe Requires Two Oscilloscope Channels and a Very Flat Signal Source